Moments of the first descending epoch for a random walk with negative drift

Abstract

We consider the first exit time $\tau =\min \{n\ge 1:S_n\le 0\}$ from the positive halfline of a random walk $S_n={\sum }_1^n{\xi }_i,n\ge 1$ with i.d.d. summands having a negative drift $\mathbb{E}\xi =-a<0$. Let ${\xi }^{+}=\max (0,{\xi }_1)$. It is well-known that, for any $c>1$, the finiteness of $\mathbb{E}({\xi }^{+}{)}^c$ implies the finiteness of $\mathbb{E}{\tau }^c$ and, for any $c>0$, the finiteness of $\mathbb{E}\exp (c{\xi }^{+})$ implies that of $\mathbb{E}\exp (c^{\prime }\tau )$ where $c^{\prime }>0$ is, in general, another constant that depends on $c$ and on the distribution of ${\xi }_1$. We consider the intermediate case, assuming that $\mathbb{E}\exp (g({\xi }^{+}))<\infty$ for a positive increasing function $g$ such that ${\mathrm{liminf}}_{x\to \infty }g(x)/\log x=\infty$ and ${\mathrm{limsup}}_{x\to \infty }g(x)/x=0$, and that $\mathbb{E}\exp (c{\xi }^{+})=\infty$, for all $c>0$. Assuming a few further technical assumptions, we show that then $\mathbb{E}\exp ((1-\epsilon )g((1-\epsilon )a\tau ))<\infty$, for any $\epsilon \in (0,1)$.

7 pages